V1 — the profile law at the caustic: c2 = 1 − (3/4)β, and Conjecture A delimited
Question. Is the deviation statistic’s leading coefficient c2 = 1 universal, or does it read the field’s profile shape beyond |∇ln B|?
Derived first (T4, period average). For B = B0 e^L with β = L’‘(0)/L’(0)²,
the θ-period average obeys c2 = 1 − β/2 (article Prop 4.1, proven at leading
order; verified at β = 0, ±1, ±2 by run_t4_beta_period_check.py).
Pre-registered prediction (V1). If conjugate time = θ-period held on every profile, the Jacobian pipeline would measure c2 = 3/2 on B = B0(1+gx).
Result: REFUTED — twice informative. run_v1_linear_profile.py
(artifacts/v1_linear_profile.json), power family B = (1+gx)^n, β = −1/n:
| profile | β | period c2 (proven) | caustic c2 (measured) | 1 − (3/4)β |
|---|---|---|---|---|
| exponential (n = ∞) | 0 | 1 | 0.9996 (P1) | 1 |
| quadratic (n = 2) | −1/2 | 5/4 | 1.3741 ± 0.0019 | 11/8 |
| linear (n = 1) | −1 | 3/2 | 1.7466 ± 0.0074 | 7/4 |
- Caustic law: c2 = 1 − (3/4)β — measured, then DERIVED. Measurement: both points
within ~half the quoted sensitivity band; the alternative “period + β²/4” misses the
n = 2 point by 0.062, ~30× that band. (Bands are fit-model spread + resolution
shift — numerical sensitivities, not sampling σ.) Derivation (
run_o6_caustic_c2.py, article Prop. 4.2): exact second-order perturbation of the Jacobian zero — τ1(θ0) = 2π sin θ0 at first order (odd ⇒ averages out), and ⟨τ2⟩ evaluates symbolically to 2π(1 − 3β/4) for an ARBITRARY one-dimensional profile jet, lifting the power-family restriction. Remaining of O6: higher coefficients and a hand-written proof. - Conjecture A (t_c = θ-period) is exponential-specific: on n = 1, 2 the per-angle gap max|t_c/T − 1| grows ∝ βε² (1.2e-2 at ε = 0.15, n = 1) while the exponential holds at 1e-8. Orbit closure + E-collapse hold on the linear profile too — so they cannot suffice to prove A.
- Critical gradient is profile-dependent: ε = 1/4 for the linear profile (radicand turning point), vs 1/2 (K’s singularity) for the exponential.
- The blind jet at β = 4/3, TESTED (F2, final review): the power profile n = −3/4 through the same pipeline gives caustic c2 = (−0.7 ± 1.4)e−5 — consistent with zero, as the law predicts, while the period average there is 1/3. The residual is consistent with ε⁴ scaling over the sampled range, under the even-power fit models used (run_f2_beta43_blind.py, artifacts/f2_beta43_blind.json; pre-registered rule: PASS).
Consequence for Part 3’s inversion. |∇ln B| = √|δ|/r_L carries a profile-calibration factor |1 − (3/4)β|^{−1/2} (absolute value: at β > 4/3 the combination is negative and δ flips sign — the β = 2 example enters that regime). Two Larmor radii do NOT separate gradient from curvature: every sufficiently small radius measures the same single combination (1 − (3/4)β)L′(0)²; the separation claim is retracted (article §4, consequence ii). Written into Part 3, D5, and article §4.
Reproduce.
cd research/preferred-directions
../cosmic-web/.venv/bin/python scripts/run_t4_beta_period_check.py
../cosmic-web/.venv/bin/python scripts/run_v1_linear_profile.py